Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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74 CHAPTER 12. THE RESIDUE THEOREM AND APPLICATIONS<br />
Pro<strong>of</strong>. Let ǫ > 0 be such that Bǫ(zj) ⊂ D for j = 1,...,n, with zk /∈ Bǫ(zj) for k �= j.<br />
For j = 1,...,n, we have Laurent representations<br />
f(z) =<br />
∞�<br />
k=−∞<br />
a (j) k<br />
k (z −zj)<br />
for z ∈ Bǫ(zj)\{zj}, so that res(f,zj) = a (j)<br />
−1. For j = 1,...,n, define<br />
hj: C\{zj} → C, z ↦→<br />
so that hj is holomorphic on C\{zj}. Define<br />
�−1<br />
k=−∞<br />
g: D \{z1,...,zn} → C, z ↦→ f(z)−<br />
a (j)<br />
k (z −zj) k ,<br />
n�<br />
hj(z),<br />
and note that z1,...,zn are removable singularities for g.<br />
Since D is simply connected, Cauchy’s Integral Theorem yields:<br />
�<br />
0 = g(γ)dζ<br />
�<br />
=<br />
�<br />
=<br />
�<br />
=<br />
�<br />
=<br />
�<br />
=<br />
γ<br />
γ<br />
γ<br />
γ<br />
γ<br />
γ<br />
f(ζ)dζ −<br />
f(ζ)dζ −<br />
f(ζ)dζ −<br />
f(ζ)dζ −<br />
f(ζ)dζ −<br />
n�<br />
�<br />
j=1<br />
n�<br />
�<br />
γ<br />
γ<br />
hj(ζ)dζ<br />
� −1<br />
�<br />
j=1 k=−∞<br />
n� �−1<br />
a<br />
j=1 k=−∞<br />
(j)<br />
k<br />
n�<br />
j=1<br />
a (j)<br />
−1<br />
�<br />
γ<br />
�<br />
j=1<br />
a (j) k<br />
k (ζ −zj)<br />
γ<br />
1<br />
dζ<br />
ζ −zj<br />
(ζ −zj) k dζ<br />
n�<br />
res(f,zj)2πiν(γ,zj).<br />
j=1<br />
Corollary 12.1.1. Let D ⊂ C be open and simply connected, f : D → C be<br />
holomorphic, and γ be a closed curve in D. Then we have<br />
ν(γ,z)f(z) = 1<br />
�<br />
f(ζ)<br />
2πi ζ −z dζ<br />
for z ∈ D\{γ}.<br />
γ<br />
�<br />
dζ